I think the translation of 出かかった means to go out, but for that tense I think it's "gone out" so I think the translation of that sentence is "Daisuke swallowed the words that had gone out through his throat"

「鷹の魂は消えないで空の中に永遠に生きる。」 "A Hawk's spirit never dies, for it lives on forever in the sky"
-Joey Vega